where \(u=u(t,x)\)  and \(\alpha \)  is a constant. If we drop the right side of the above equation, then we recover the BBM equation, which was proposed by Benjamin, Bona and Mahony [1]} as an alternative to the KdV [2]} equation for describing waves in water models. However, both KdV and BBM equations do not describe wave breaking, which is a common, but mathematically challenging, phenomenon arising in waves propagating in shallow water regimes.
